Learning Out Loudhttps://jenniferbrokofsky.wordpress.com
The adventures of a teacher and mother who is passionate about learning mathematics and sharing that learning with others. Mon, 24 Apr 2017 20:31:30 +0000en
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Push In Posters for Early Mathematicshttps://jenniferbrokofsky.wordpress.com/2017/04/24/push-in-posters-for-mathematics/
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As students are playing and exploring mathematics there are times when teachers can step back, watch, and document the learning. These moments allow teachers to assess the students understanding of the concepts and formulate ideas for future instruction. At other times students may require teachers to “push in” to the conversations and explorations to support students in making connections, taking next steps, and engaging in mathematical conversations. Pushing in may look like a teacher pulling up a chair and joining the group or placing a support in front of students to direct the conversations and the learning. One tool for this support can be Push in Posters. These posters can put beside students to help them assess vocabulary, or make a connection. What I like about using them is that the support right beside the learners and not across the room like a Math Wall may be. To support your young mathematicians I have created Push In Posters for Number Recognition, Patterns, and Shapes in both English and French. I hope that you find them helpful.

]]>https://jenniferbrokofsky.wordpress.com/2017/04/24/push-in-posters-for-mathematics/feed/0IMG_1662jenniferbrokofskySubitizing- More than Just Dotshttps://jenniferbrokofsky.wordpress.com/2017/04/16/subitizing-more-than-just-dots/
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Subitizing is the ability to quickly identify the number of items in a small set without counting. To do this effectively students are required to “see” collections or groups as opposed to counting one at a time. They need to see a configuration of objects and instantly know how many is there. This skill requires practice and support. The practice can come from things such as dot cards, ten frame cards etc. In these cases students can be shown the card quickly and then asked the question “How many did you see?” The support comes from the conversations that ensue. As students are asked to articulate their thinking about what they saw and how it supported them in knowing “how many” their thinking grows. As students hear from others how they “saw” the collection their thinking is supported.

The value of subitizing with your students is considerable. Research indicates that subitizing can play an important role inthe development of mathematics skills including addition/subtraction, and multiplication/division. It can also support students development in number sense and can serve as an indicator for student success in mathematics.

Our curriculum has specific subitizing outcomes for Kindergarten and Grade 1 students. However, the value of this work can extend well beyond the primary grades. Subitizing activities can serve as a powerful warm up or Minds On activity to start a mathematics class at any level.

To support you with subitizing with your young mathematicians check out these Subitizing Cards.

]]>https://jenniferbrokofsky.wordpress.com/2017/04/16/subitizing-more-than-just-dots/feed/0SubitizingjenniferbrokofskyWhat Stories Live in These Equations?https://jenniferbrokofsky.wordpress.com/2017/02/24/what-stories-live-in-these-equations/
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Learning is embedded in memory, history and story.

The human brain is wired to remember stories. In many cultures, the power of storytelling has been harnessed to teach important lessons, pass down traditions, and make sense of the world. Storifying content can help culturally diverse learners (and all learners) build contexts for learning, identify similarities and differences, find relationships, notice how concepts fit together, and recognize a different point of view. This is the brain’s way of weaving it all together and making learning stick.

At our November K-5 Mathematics Leadership Community we began our time together with a provocation at each table that invited teachers to them to choose an equation from a pile of possibilities and create a story to go with it. I had seen this idea in a tweet from Janice Novakowski and was inspired to recreate it for our learners. The equations ranged from grade 1 to grade 6 content, and had the unknown part of the equation in different places within the equations. This storifying provocation provoked learners to:

develop their own contexts for equation of their choosing

share stories with others

engage in conversations about creating problems and problem types

connect math to familiar contexts

This provocation also allowed us, as teachers to dig into Cognitively Guided Instruction (CGI). CGI is a research based approach to teaching mathematics that builds on children’s natural problem solving strategies within different contexts. CGI allows teachers to identify problem types which students are comfortable working with and ones they may not be using or working with…yet. Through identification teachers can support student learning of all problem types and applications.

At our last math community meeting teachers were given an opportunity to explore the provocation shown above. This provocation was provided to support wonder, curiosity, creativity and multiple opportunities for computation. Teachers were able to create designs with addition, and multiplication using one of my favorite computation manipulatives…Cuisenaire rods.

I chose this activity and these manipulatives for many reasons.

It allows students to explore part-part-whole thinking

It provides opportunities for multiple computations in an engaging way

It brings creativity, and design into mathematics

It allows for collaboration, and conversation

Students can discover relationship, and multiple interpretations

After our professional exploration many teachers took this activity back to their classrooms and shared it with their young mathematicians. Below are some examples of what the students came up with in classrooms.

As a follow up students can bridge the concrete to abstract continuum by recording their computations/designs pictorially and abstractly on a sheet. Initially, recording sheets could have pictures of coloured rods to match the concrete representations the students created. As students progress in their level of abstraction they can switch to just marking the value on the rod. Abstractly, they would record the equation their design represents.

Below are some supports that can help you incorporate Cuisenaire rods into your classroom.

A Mathematical Clothesline is a great visual tool to support students in reasoning proportionally with numbers. Students can work cooperatively to place a set of concept cards (various representations of number) on the line in a position that makes sense proportionally. As students place the cards on the line they need to consider the card they have, the cards left to position, and any benchmark numbers (numbers that assist in estimation and tend to be multiples of 5 or 10) that are already present. Mathematical Clotheslines differ from a formal number line in that cards are not placed in a measured and scaled sequence, but simply displayed in a specific order.

I have seen many clothes line activities online in recent months but only recently this activity for younger students. Janice Novakowski recently created this post which inspired me to explore her idea to use this tool with younger students. I expanded on her idea by adding in number words in English and French as indicated by our grade one curriculum outcome. I have also created a fraction version.

In both of the Mathematical Clotheslines below students have multiple representations of numbers to work with. Not all representations need to be introduced and used at the same time. As students expand on their understanding of different ways to represent numbers and fractions pictorially more and more representations can be added to the activity and become a topic of discussion in the classroom.

The word routine is defined as a sequence of actions that are regularly followed so that they become a habit. In the classroom routines are often thought of as classroom management procedures that support students in making more effect use of classroom time and to transition more efficiently from one activity to another. Classroom routines such as how to sharpen your pencil, clean up your materials, come in from recess etc. are explicitly taught, and practiced early on in the school year. As these routines become habits they help the classroom community run more smoothly, save valuable learning time, and create a predicable, safe space for learners and learning.

Routines can also be powerful tools for mathematical thinking and reasoning. “Like the management routines, these “mathematical thinking routines” also have a predictable set of actions that students learn and then practice repeatedly until they are second nature.” (Kelemanik, Lucennta, and Creighton, 2016, p.18) Number talks are one such reasoning routine. As you use Number Talks in your classroom you establish a routine of actions which the students come to expect and become familiar with. These actions form a habit and in so doing free up cognitive energy so that the brain can focus not on the routine but on the mathematics.

Think of a routine as a container. The container does not change, it is a constant, it is solid, it is reliable. You can count on the container to hold what you need it to hold. When the container is well established for students their cognitive energy can shift from focusing on the container to focusing on the contents in the container. This is the power of an established and predictable Number Talk routine.

The routine of a Number Talks has 5 main steps that students come to expect. During a Number Talk they will:

View a problem/ task.

Think and solve it.

Share their answer.

Defend their solution.

Compare and connect their reasoning to the reasoning of others.

The benefits of adding this promising practice to your instructional routines are numerous. Besides building more confident mathematicians, Number Talks increase students number sense as they require students to become more flexible and strategic thinkers. Number Talks also, increase students ability to articulate their thinking and refine their mathematical communication skills. As we work to increase our students computational fluency Number Talks have become one of THE most promising practices we can add to your professional repertoire.

If you would like to learn more about other Routines for Reasoning check out these books:

Provocation can be defined as the act of causing someone to feel, think or begin to do something. In the Reggio Emilia approach to learning they involve the teacher creating invitations/ displays that provoke students to begin to explore an idea or concept with materials that spark thinking. Provocation materials can be loose parts, natural items, children’s literature, photographs, inquiry questions or all of the above. The goal is to find something that sparks curiosity and leads to new learning.

Within mathematics, provocations can lead to inquiry and discovery of mathematical concepts and initiate mathematical thinking. I am currently working with a group of four exceptional kindergarten teachers to develop opportunities for our elementary teachers to explore the idea of using provocations as a tool to provoke mathematical inquiry, creativity, and discussions. Together we initiated the first ever Kindergarten Mathematics Learning Community in which kindergarten teachers come together to discuss ways to provoke their young mathematicians in thinking and growing mathematically.

Our first provocation focused on the question How do Numbers Help Me Tell My Story? This question and the materials we deliberately put together can serve as a starting point for developing a deeper understanding of numbers and counting concepts in Kindergarten.

Some of the Big Ideas of Counting that can be explored through this provocation are:

You say one and only one number for each object (one to one correspondence)

The last number spoken tells how many. (cardinal principle)

You can represent a number in a variety of ways.

The quantity of a group does not change if the objects are rearranged. (stability)

There is a consistent set of counting words that never changes. (stable order principle)

You say one number name for each object tagged. (synchrony)

Our How do numbers help me tell my story? provocation included a variety of loose parts (shells, buttons, rocks, glass beards, pipe cleaners etc.) as well as tree cookies with numbers printed on them and wooden numbers. We used felt squares to delineate the space and added round cork pot holders. Looking in the mirror students could study themselves to recreate what they see using the loose parts. They can also think about numbers that connect to themselves and represent those numbers using the loose parts on the counting place mats as a guide.

Using high quality children’s literature to connect mathematical concepts, and cultural perspectives is an important consideration for us in every provocation we design. The Colors of Us was our inspiration for our How do numbers help me tell my story? provocation. Other literature connections can include The Best Part of Me, and Every Buddy Counts.

For those of you wanting to provoke the thinking of our young mathematicians I am including a freebie to get you started. Here are the counting mats I created in French and in English as well as the Number Formation Cards. Enjoy!!

This exploration of Provocations in mathematics is has become a wonderful learning journey for me. I am so honored to have an opportunity to learn with some fantastic teachers within my school division and around the world. If you are looking for inspirations from leaders in this area make sure to check out the work of Janice Novakowski.

]]>https://jenniferbrokofsky.wordpress.com/2016/11/19/provoking-exploration-in-mathematics/feed/0provokationsjenniferbrokofskyimg_0283img_0284Getting Started with Mathematician’s Workshophttps://jenniferbrokofsky.wordpress.com/2015/09/30/getting-started-with-mathematicians-workshop/
https://jenniferbrokofsky.wordpress.com/2015/09/30/getting-started-with-mathematicians-workshop/#respondWed, 30 Sep 2015 20:45:58 +0000http://jenniferbrokofsky.wordpress.com/?p=1163Continue reading Getting Started with Mathematician’s Workshop]]>At the beginning of the school year I was fortunate enough to work with many dedicated teachers around the idea of Mathematician’s Workshop. Our day together revolved around a framework I had created as a consolidation of my learning about successful mathematics instruction in the classroom. This is a synthesis of many ideas from many mathematics educators and experts.

To explain my thinking of this model and to provide an example of how it is working in a classroom I was fortunate enough to have an opportunity to create a Periscope with one of my session participants. Mrs. Colleen Johnson was so very generous with her time and expertise to share how she is making Mathematician’s Workshop live in here grade 4/5 classroom.

I have spent the last 3 days listening to, learning from, and devouring the words of great teachers. Debbie Miller, Patrick Allan, and Penny Kittle have reminded me of the bliss that comes from teaching students how to read and write using authentic experiences with reading and writing. Authentic moments where you connect the students to their inner reader by helping them find the texts that matter to them and engage them in ways that only powerful text can. Authentic moments that connect them to their inner writer by standing on the shoulders of “beautiful words” from others like Sarah Kay, and writing about things that evoke their passions. If I could take away one word…one message…one thought from these last three days it would be Authentic!!!

For these great teachers- Debbie, Patrick, and Penny authentic is about providing students with opportunities to engage in reading and writing in ways that are meaningful, interesting, and relevant to students as individuals. In their classrooms they provide students with time to connect with great books while they take the time to confer with students. In writing they provide students with time to write as they themselves share their own writing both finished products and works in progress. This kind of teaching is not born from worksheets, packaged programs, and activities that do nothing more that fill the time. It comes from relationships, sitting side by side with students, and taking to them about their lives as readers and writers.

These 3 days have provided me with time to think about what this could look like in mathematics. Can we take these ideas and put them into our mathematics classrooms? Can we help students connect as mathematicians who learn from the “beautiful” math of others? Can we model our own mathematics work and how we engage in mathematics? Can we provide students with time to actively engage in rich authentic mathematics that provokes thinking, and passion?

This type of mathematics is about more than completing the worksheet, or the page in the textbook. It is about fostering the conditions for student to become mathematicians who don’t just do math but think mathematically. My mind is swimming with the possibilities of re-imagining what the Mathematics Workshop could look like so that it would foster mathematicians. I think it is possible.

One thing is clear though. The most important thing that we as teachers need is an unwavering belief that all kids are capable of becoming mathematicians and deserve the time to dig into authentic mathematical experiences.

]]>https://jenniferbrokofsky.wordpress.com/2015/06/10/authentic-learning-connecting-literacy-to-mathematics/feed/3Authentic LearningjenniferbrokofskyAuthentic LearningIf You Give a Kid Some Cards They Will…#PlayMathhttps://jenniferbrokofsky.wordpress.com/2015/05/17/if-you-give-a-kid-some-cards-they-will-playmath/
https://jenniferbrokofsky.wordpress.com/2015/05/17/if-you-give-a-kid-some-cards-they-will-playmath/#commentsMon, 18 May 2015 05:28:33 +0000http://jenniferbrokofsky.wordpress.com/?p=1135Continue reading If You Give a Kid Some Cards They Will…#PlayMath]]>I love to play cards. It is a love that has been instilled in me by my parents and grandparents. Now I am lucky enough to have my kids picking up this love from my family. The joy for me comes not only from the game but from the memories of get seeing 4 generations of my family sitting around a table sharing time together, conversations, and fun. My kids have 3 favorite games that they have been taught by their great grandparents. They ask to play them every time we are all together. These games are great for the kids because they are easy to learn and also allow them to use their ever growing skills in mathematics. I wanted to share them with you. Maybe they will become an opportunity for you to sit around a table with your own children and #PlayMath.

Chase the Ace

The object of the game is to not not be stuck with the lowest card. In this game the kings are high and aces are low. To play this game you need a standard set of cards and three counters per person (nickles, paperclips, buttons…). The dealer deals out one card to each player. Each player then looks at their card and determines whether or not they want to keep it or trade it with the person to their left. The player to the left of the dealer starts by saying “keep” if they want to keep the card or trades it with the person to their left by sliding it face down to the player. That player must then exchange cards with the player wanting to change unless they have a king. If they have a king they lay the king down face up and show the rest of the players as proof. Having the king makes them immune to trades for that round. If that happens the trade can not occur and the player must keep the card. The big risk comes in when changing your card because you always run the risk of changing for a lower card. After the exchange has taken place the player who was forced to trade looks a their card. They then decide if they want to stand or change with the player to their left. Play continues around the table until it reaches the dealer. If the dealer wants to exchange cards they turn theirs up and cut the deck. The card they cut becomes their card.

Once the play is back to the dealer all players lay their card down face up for all to see. The player with the lowest card places one of their counters into the pot. If more than one player has the lowest card, each player with the low card must place a counter into the pot. The cards are collected to be shuffled and dealt by the next player to the left of the dealer. In this way the role of the dealer also goes around the table.

Once a player loses all their counters they are out of the game. The winner is the last player to still have a counter.

Sticks- A Game of Sets and Runs

This is the current favorite around the table. All you need to play is a set of large Popsicle sticks (we bought ours from the dollar store) and a deck of cards. My mom actually made 2 sets. One that is more challenging for us adults, and a junior version for the kids. In this way everyone is playing together. The object of this game is to successfully accomplish the task on your stick before someone around the table goes out. To go out you are able to get rid of all of your cards by laying out your own sets and runs or by playing on the sets and runs of others. Each time you accomplish a task you “earn” your stick and can turn it face up. The first person to earn seven sticks wins.